I just finished watching the show Devs. A big part of the show is using a computer to accurately view events in the past, and there is a lot of discussion about free will versus determinism.
If free will exists, it would be impossible for a computer to predict the future. However, would it be possible to reconstruct the past since the computer would know the result of every decision a person made?
The essence of “chaos theory” — the observation that we have sensitive dependence on initial conditions in most causal behavior — is that the sheer volume of necessary data input and processing necessary to do precision projection would always require a computer that consisted of more matter and more energy than the entirety of energy and matter available in the specified environment.
Thus, if by “world” you mean the entirety of universe that could conceivably have an effect that would interact with Abe’s hat, the computer would need to exist in a much larger universe capable of studying the Abe-hat-containing world like a little laboratory experiment.
In essence, any given closed system will have more chaos than it is capable of studying.
If we allow for an infinite amount of compute power, and thus side step the ill conditioned nature of the calculation, we get to the question of the arrow of time.
Physical processes are reversible, and it is an interesting problem to come up with a physical reason for the direction of time anyway. Questions of entropy and thermodynamics are usually invoked.
Now we get to free will.
However this raises a similar problem. Whilst we know the result of the decision, how do we deduce the question? If free will is involved, the result does not necessarily tell us that. So the problem is likely symmetric, and similarly not possible.
Not sure if I should answer the OP or the title.
You cannot know the position of every particle to within Planck’s length. And you also have uncertainty in velocity so even if you know where a particle is, you cannot say where it was or will be.
Uncertainly certainly comes into play. Increasing exactness of knowledge of position requires decreasing knowledge of momentum. Perfect knowledge, i.e. infinite precision, of position requires zero knowledge of momentum (not velocity), and zeroes can’t be allowed in the equation, so perfect knowledge is unattainable.
Approximations also have their problems. Each added decimal point of precision is harder to calculate and so adds to the time of processing. Wiki gives the number of atoms in the world as 133,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000. Calculating the position of each out to many decimal points would require deep time. You aren’t going to get your answer back by the end of the show.
Wouldn’t Lincoln have taken off his hat indoors?
If he didn’t, Booth would have taken it off for him.
No.
Stranger
That is what I thought. All the illustrations show him hat-less.
True, but the title just asks the position of the hat. Surely he had it with him.
Little known fact: He was wearing his tam o’shanter that night.
Free will or no free will, fundamentally complete information is not knowable.
First there is Heisenbergs principle. Knowing position completely and speed completely are in contradiction.
Second is quantum froth. If quantum theory is correct then randomness is inherent at the smallest levels.
Could you put that in layman’s terms? If I look at a pool table there is no way to know what the configuration of the balls was before the last shot. Isn’t that a bit like the Lincoln’s hat question?
At the macro level that’s true, it seems that you lose the information of where the balls on the pool table were previously. At the particle scale though, the information is still there and there is no fundamental difference between calculating forwards in time or backwards in time.
If this is a hijack I’ll gladly start a new thread but I’m not following this. I took two semesters of classical mechanics and a semester of quantum mechanics but I’m no physicist and that was 40 years ago.
So how is that information still there? I would think you could determine the energy at the particle scale but not direction. They didn’t keep their ticket stubs.
It’s beyond me too. @Francis_Vaughan or one of the other more knowledgable types would be better to handle the finer details.
I’m not a physicist either. But the problem is more a philosophical one anyway. The usual setting of such questions assumes a-priori that full knowledge of the state of the system is known. Questions about whether this is possible are another matter. But are important in the wider context. Similarly we are concerned with the underlying principle, and can posit any amount of compute power we like, and let it run as long as we need. Infinite compute is a good start.
Currently we believe that information cannot be destroyed. Black holes being ignored for the moment. Quantum processes can run either direction, and it isn’t intrinsically possible to say which way they are going. Feynman famously realised that his eponymous diagrams worked no matter which orientation they had.
So the past is in some sense part of the present.
Free will versus the clockwork universe appears about now.
We can sweep the issue away by claiming true random processes inside quantum mechanics. But that is of course only one of the various possibilities. Hidden variables, pilot waves, super determinism and their ilk plus splitting universes are also available. As are attempts to shoehorn consciousness and free will into the discussion. Penrose being the most famous proponent.
The OP’s question seems to be a clever way of reintroducing the argument in reverse.
Usually we sweep up the question of free will and choice with all possible variations of interpretations of QM.
The reverse time argument seems to lean on an ideal that forward running time results in more possible outcomes treeing out from the past, and so fewer in the past. However I don’t think you can do this. Total entropy of the universe may be lower, but the state space, and information content needed to describe it is no less. As Douglas Adams realised, even if you have the answer, the question may still be unknown.
Hope this made sense. I typed it on my phone waiting for a friend to go to the opera. Mimi gets to die of tuberculosis again. 
I think the answer has to tend towards “yes”, given a bit of vagueness in the question and given an “all powerful computer and knowledge of every particle…”.
I also think “free will” sounds like it means something very explicit, but it actually doesn’t. And that this is irrelevant in trying to, if you will, predict the past.
It’s awfully, awfully hard to predict the weather. But in just one century, look how much better it’s been made. And what’s one teeny little century, in the grand sweep of time? How much more powerful will computers be in a million years? In a billion? “All powerful” is better than that. And reconstructing details of past weather systems is surely easier than future ones.
Seems beside the point to speculate how powerful real computers may be, when you are already talking about infinitely fast computers processing an infinite amount of (IRL unobservable) information.
Basically, if you know the dynamics of a system (its laws of evolution), and its (quantum) state at any one time, you know the system’s state at any given time. This is a direct consequence of what’s already been said, namely, that physical laws are information-preserving—so the information contained in the state at any given time is the complete information about the system. So, in principle, yes: you could calculate the exact prior state.
The question is, though, what that would tell you. Let’s take the example of a single particle. Its position and velocity (momentum) aren’t simultaneously perfectly precise—rather, both are definite up to some degree of accuracy (a pretty high degree of accuracy, actually, which is why we don’t usually have to deal with this sort of thing). The extreme points of this is that the particle has a perfectly defined position, and completely undefined velocity, or the other way around—but these are actually sort of pathological cases, and you have to employ some mathematical tricks to even work with them (such states aren’t normalized, and you have to ‘rig’ your Hilbert space to account for them).
So the complete quantum state of the system will be one where the particle has position and momentum definite up to some certain accuracy. And under the system’s dynamics, that’ll evolve to a state with a (generally different) accuracy for each of position and momentum—it could grow more or less localized, for instance. Knowing that state at any one point in time, and knowing the dynamics, one could predict how localized the particle is at any given point in time—i. e. give the likelihood of finding it at any given point, with ‘perfect’ localization amounting to certainty of finding it at some given point.
But this produces two challenges. Suppose the question you’re interested in is: where was the particle at 10 o’clock yesterday evening? Well, you can only calculate the probability distribution of where you might’ve found it, had you looked. That’s the general problem with quantum states: even in the present, unless the system is in a state of perfect localization, a quantum state can’t give you the answer to where you’ll find the system; only measurement can do that, and quantum states only provide probabilities of certain measurement outcomes.
This leads to the second challenge. How do you know the quantum state of a particle now? If, for instance, you try to measure its position, after the measurement, you’ll certainly have a quantum state describing a particle with a highly localized position. But that doesn’t entail that the particle was highly localized before—so your measurement doesn’t tell you what the quantum state of the system was, but only, what it now is. To perfectly measure the quantum state of a particle, you need in general many copies of it, and perform a process known as quantum tomography—repeating measurements a great number of times to obtain the probability distributions for different outcomes. (The only other way is to know beforehand which of a set of states you might expect, and perform a measurement in that basis.)
So, the trouble with ‘reconstructing’ the past state of the universe isn’t free will or loss of information, or computational power or the like, but simply that even the present state doesn’t generally give us unambiguous answers for the questions we might want to ask, and furthermore, given that we only have one universe to experiment on, itself is probably unknowable.
So third: Gödel. Or at least GEB. At least the fundamental necessity of incomplete information.
The universe is the set whose complete information is contained within this computer which contains within it the set that is all of itself as one member of the set it is studying. No matter how infinitely large it cannot keep up.